## Noise in Nonlinear Dynamical Systems: Volume 1, Theory of Continuous Fokker-Planck SystemsNature is inherently noisy and nonlinear. It is noisy in the sense that all macroscopic systems are subject to the fluctuations of their environments and also to internal fluctuations. It is nonlinear in the sense that the restoring force on a system displaced from equilibrium does not usually vary linearly with the size of the displacement. To calculate the properties of stochastic (noisy) nonlinear systems is in general extremely difficult, although considerable progress has been made in the past. The three volumes that make up Noise in Nonlinear Dynamical Systems comprise a collection of specially written authoritative reviews on all aspects of the subject, representative of all the major practitioners in the field. The first volume deals with the basic theory of stochastic nonlinear systems. It includes an historical overview of the origins of the field, chapters covering some developed theoretical techniques for the study of coloured noise, and the first English-language translation of the landmark 1933 paper by Pontriagin, Andronov and Vitt. |

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### Contents

Some Markov methods in the theory of stochastic processes | 16 |

Langevin equations with colored noise J m sancho | 72 |

First passage time problems for nonMarkovian processes | 110 |

applica | 161 |

Methods for solving FokkerPlanck equations with applications | 191 |

Macroscopic potentials bifurcations and noise in dissipative | 225 |

Transition phenomena in multidimensional systems models | 279 |

### Common terms and phrases

applied approximation assume attractor BFPE bifurcation bistable boundary conditions Brownian motion calculation coefficients colored noise consider const constant correlation function corresponding defined deterministic diffusion function discussed dynamical systems EFPE eigenvalues equilibrium equilibrium thermodynamics example exponential Faetti Figure fluctuations Fokker-Planck equation Fronzoni functional derivative functional probability density Graham Grigolini Haken Hanggi Hernandez-Machado initial condition integral Jung Kampen Kramers Landauer Langevin equation Lett limit cycle Lindenberg and West linear Lorenz model Mannella Marchesoni Markov Markov process Markovian Masoliver matrix McClintock mean first passage method Miguel and Sancho Moss non-Markovian nonequilibrium potential nonlinear obtain oscillator parameter perturbation phase space Phys physical problem properties random result right-hand side Risken San Miguel Schenzle Section separatrices solved Springer Stat stationary probability statistical stochastic differential equation Stochastic Processes Stratonovich techniques theory thermodynamics trajectory transformed transition values variables white noise zero